PL-OM 1988

The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$ , where $m$ is an integer and $r \in [0,1)$ . Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$ .

For a permutation $P = (p_1, p_2, ... , p_n)$ of $(1, 2, ... , n)$ define $X(P)$ as the number of $j$ such that $p_i < p_j$ for every $i < j$ . What is the expected value of $X(P)$ if each permutation is equally likely?

$W$ is a polygon which has a center of symmetry $S$ such that if $P$ belongs to $W$ , then so does $P'$ , where $S$ is the midpoint of $PP'$ . Show that there is a parallelogram $V$ containing $W$ such that the midpoint of each side of $V$ lies on the border of $W$ .

$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$ . Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$ .

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$ , $a_{n+3} = a_{n+2}a_{n+1} + a_n$ . Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$ .

Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$ .